Numerical evidence on the uniform distribution of power residues for elliptic curves

Abstract

Elliptic curves are fascinating mathematical objects which occupy the intersection of number theory, algebra, and geometry. An elliptic curve is an algebraic variety upon which an abelian group structure can be imposed. By considering the ring of endomorphisms of an elliptic curve, a property called complex multiplication may be defined, which some elliptic curves possess while others do not. Given an elliptic curve $E$ and a prime $p$, denote by $N_p$ the number of points on $E$ over the finite field ${\mathbb{F}}_p$. It has been conjectured that given an elliptic curve $E$ without complex multiplication and any modulus $M$, the primes for which $N_p$ is a square modulo $p$ are uniformly distributed among the residue classes modulo $M$. This paper offers numerical evidence in support of this conjecture.